Following [1]}, [2]}, [3]}, for fixed \( r\in [0,1], \)  \( t>0 \)  and integer \( N>0 \) , we define the compact subset of \( \Omega \times \mathbb {R}_+ \) ,
\(\Omega _{(r,t)}^N & = \lbrace  (x, \tau ): \Gamma _{r,-}^N(\tau ) < x_1 < \Gamma _{r,+}^N(\tau ), x^{\perp }\in \mathbb {T}^2, 0<\tau <t \rbrace  \\\text{with} \quad \Gamma _{r,\pm }^N(\tau ) & := r+ s\tau + Y(\tau ) \pm N.\) 
